The value of of sum of the series `""^(14)C_(0).""^(15)C_(1)+""^(14)C_(1).""^(15)C_(2)+…+""^(14)C_(14).""^(15)C_(15)` is

Options

A. `""^(29)C_(10)`
B. `""^(29)C_(12)`
C. `""^(29)C_(14)`
D. `""^(29)C_(18)`

Answer

To find the value of the sum of the series

\[ \sum_{k=0}^{14} \binom{14}{k} \binom{15}{k+1} \]

we can use a combinatorial identity. The identity we will use is:

\[ \sum_{k=0}^{n} \binom{m}{k} \binom{n}{r-k} = \binom{m+n}{r} \]

In our case, we can rewrite the series as follows:

1. **Identify the parameters**: Here, \( m = 14 \), \( n = 15 \), and \( r = 15 \).
2. **Adjust the indices**: Notice that we can express the sum in a way that fits the identity by changing the index of summation. We can rewrite \( \binom{15}{k+1} \) as \( \binom{15}{r-k} \) where \( r = 15 \) and \( k \) goes from \( 0 \) to \( 14 \).

Thus, we can rewrite the sum as:

\[ \sum_{k=0}^{14} \binom{14}{k} \binom{15}{15-k} = \binom{14+15}{15} \]

3. **Calculate the binomial coefficient**: Now we compute:

\[ \binom{29}{15} \]

This gives us the value of the sum.

4. **Final answer**: Therefore, the value of the sum of the series is:

\[ \binom{29}{15} \]